SARAJEVO JOURNAL OF MATHEMATICS
Vol.5 (18) (2009), 211–233
BASINS OF ATTRACTION OF EQUILIBRIUM POINTS OF
MONOTONE DIFFERENCE EQUATIONS
A. BRETT AND M. R. S. KULENOVIĆ
Dedicated to Professor Harry Miller on the occasion of his 70th birthday
Abstract. We investigate the global character of the difference equation of the form
xn+1 = f (xn , xn−1 , . . . , xn−k+1 ),
n = 0, 1, . . .
with several equilibrium points, where f is increasing in all its variables.
We show that a considerable number of well known difference equations
can be embeded into this equation through the iteration process. We
also show that a negative feedback condition can be used to determine
a part of the basin of attraction of different equilibrium points, and that
the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of
neighboring saddle or non-hyperbolic equilibrium points.
1. Introduction
Let I be some interval of real numbers and let f ∈ C 1 [I × I, I]. Let
x̄1 , x̄2 ∈ I, 0 ≤ x̄1 < x̄2 be two equilibrium points of the difference equation
xn+1 = f (xn , xn−1 ),
n = 0, 1, . . .
(1)
where f is a continuous and increasing function in both variables. There are
several global attractivity results for Eq.(1) which give the sufficient conditions for all solutions to approach a unique equilibrium. These results were
used efficiently in monograph [15] to study the global behavior of solutions
of second order linear fractional difference equation of the form
α + βxn + γxn−1
xn+1 =
, n = 0, 1, . . .
(2)
A + Bxn + Cxn−1
with non-negative initial conditions and parameters.
2000 Mathematics Subject Classification. 39A10, 39A11.
Key words and phrases. Basin, difference equation, attractivity, invariant manifolds,
periodic solutions.
212
A. BRETT AND M. R. S. KULENOVIĆ
We list three such results:
The first theorem, which has also been very useful in applications to
mathematical biology, see [13], was really motivated by a problem in [9].
Theorem 1. ([8]. See also [9, 13], and [14], p. 53). Let I ⊆ [0, ∞) be
some interval and assume that f ∈ C[I × I, (0, ∞)] satisfies the following
conditions:
(i) f (x, y) is non-decreasing in each of its arguments;
(ii) Eq.(1) has a unique positive equilibrium point x ∈ I and the function
f (x, x) satisfies the negative feedback condition
(x − x)(f (x, x) − x) < 0
for every
x ∈ I − {x}.
(3)
Then every positive solution of Eq.(1) with initial conditions in I converges
to x.
The second result was obtained in [15] and it was extended to the case of
higher order difference equations and systems in [19, 26, 32].
Theorem 2. Let [a, b] be an interval of real numbers and assume that
f : [a, b] × [a, b] → [a, b]
is a continuous function satisfying the following properties:
(a) f (x, y) is non-decreasing in each of its arguments;
(b) Eq.(1) has a unique equilibrium x ∈ [a, b].
Then every solution of Eq.(1) converges to x.
The following result has been obtained recently in [1].
Theorem 3. Let I ⊆ R and let f ∈ C[I × I, I] be a function which increases in both variables. Then for every solution of Eq.(1) the subsequences
∞
{x2n }∞
n=0 and {x2n+1 }n=−1 of even and odd terms of the solution do exactly
one of the following:
(i) Eventually they are both monotonically increasing.
(ii) Eventually they are both monotonically decreasing.
(iii) One of them is monotonically increasing and the other is monotonically decreasing.
Remark 1. Theorem 1 is actually a special case of Theorem 2. Indeed (3)
implies that there exist a and b, a < x̄ < b such that f (a, a) > a, f (b, b) < b,
which in view of monotonicity of f implies that
f : [a, b] × [a, b] → [a, b],
and so all conditions of Theorem 2 are satisfied. Furthermore, Theorem 2 is a
special case of Theorem 3 if we additionally assume non-existence of periodtwo solutions. None of these results provide any information about the
MONOTONE DIFFERENCE EQUATIONS
213
basins of attraction of different equilibrium points when there exist several
equilibrium points, as in the case of equations (4) and (21).
Example 1. Here we consider the following equation
1
xn+1 = (xn + xn−1 + sin(xn ) + sin(xn−1 )), n = 0, 1, . . .
(4)
2
on an interval [0, 2Kπ], where K is an integer. This equation has 2K + 1
equilibrium points x̄m = mπ, m = 0, 1, . . . , 2K. An immediate checking
shows that x̄2m+1 are locally asymptotically√ stable, while x̄2m are saddle
equilibrium points with eigenvalues λ± = 1±2 5 . The expression f (x, x) − x,
where f (x, y) = 12 (x + y + sin(x) + sin(y)) becomes
f (x, x) − x = sin x < 0
if
x ∈ ((2m + 1)π, (2m + 2)π).
This shows that the negative feedback condition is satisfied in the interval
(x̄2m+1 , x̄2m+2 ) and so in view of Theorem 1 or Theorem 2 the product of
this interval by itself is a part of the basin of attraction of x̄2m+1 . Similarly (x̄2m , x̄2m+1 )2 is a part of the basin of attraction of x̄2m+1 . In view of
Theorem 3 all solutions of Eq.(4) are eventually monotonic and so are convergent. Notice that Eq.(4) has no minimal period-two solutions. A direct
calculation shows that the eigenvectors which correspond to two eigenvalues
λ± at the equilibrium points (x̄2m , x̄2m ) are [1, λ± ].
The problem for Eq.(4) is to determine precisely the basin of attraction
of all local attractors x̄2m+1 . By using software that simulates discrete
dynamical systems, such as Dynamica 2, see [17], we obtain the basins of
attraction depicted in Figure 1.
Now, using Theorem 14 which is the main result of this paper, we obtain
the precise description of the basin of attraction B2m+1 of x̄2m+1 and B2m
of x̄2m as:
B2m+1 = {(x, y) : ∃yu , yl : yl < y < yu
(x, yl ) ∈ W2m , (x, yu ) ∈ W2m+2 }
and
B2m = W2m ,
where W2m = W(x̄2m ,x̄2m ) is the global stable manifold of the equilibrium
point (x̄2m , x̄2m ). In other words, the basin of attraction of (x̄2m+1 , x̄2m+1 )
is the set of all points in the plane of initial conditions which are between
the stable manifolds of two consecutive saddle equilibrium points (x̄2m , x̄2m )
and (x̄2m+2 , x̄2m+2 ).
These results are summarized in the plot in Figure 1.
A. BRETT AND M. R. S. KULENOVIĆ
214
15
W4
(x̄3 , x̄3 )
10
B3
(x̄2 , x̄2 )
5
W2
(x̄1 , x̄1 )
0
(x̄0 , x̄0 )
B1
W0
-5
-10
-10
-5
0
5
10
15
Figure 1: Figure shows the basins of attraction of Eq.(4). Figure was generated by Dynamica 3 [17].
2. Preliminaries
We now give some basic notions about fixed points and monotonic maps
in the plane.
Consider a map T on a nonempty set S ⊂ R2 , and let e ∈ S. The point
e ∈ S is called a fixed point if T (e) = e. An isolated fixed point is a fixed
point that has a neighborhood with no other fixed points in it. A fixed
point e ∈ S is an attractor if there exists a neighborhood U of e such that
T n (x) → e as n → ∞ for x ∈ U; the basin of attraction is the set of all
x ∈ S such that T n (x) → e as n → ∞. A fixed point e is a global attractor
on a set K if e is an attractor and K is a subset of the basin of attraction
of e. If T is differentiable at a fixed point e, and if the Jacobian JT (e) has
one eigenvalue with modulus less than one and a second eigenvalue with
modulus greater than one, e is said to be a saddle. See [27] for additional
definitions (stable and unstable manifolds, asymptotic stability).
Consider a partial ordering ¹ on R2 . Two points v, w ∈ R2 are said to
be related if v ¹ w or w ¹ v. Also, a strict inequality between points may
MONOTONE DIFFERENCE EQUATIONS
215
be defined as v ≺ w if v ¹ w and v 6= w. A stronger inequality may be
defined as v = (v1 , v2 ) ¿ w = (w1 , w2 ) if v ¹ w with v1 6= w1 and v2 6= w2 .
For u, v in R2 , the order interval Ju, vK is the set of all x ∈ R2 such that
u ¹ x ¹ v.
A map T on a nonempty set S ⊂ R2 is a continuous function T : S → S.
The map T is monotone if v ¹ w implies T (v) ¹ T (w) for all v, w ∈ S,
and it is strongly monotone on S if v ≺ w implies that T (v) ¿ T (w) for
all v, w ∈ S. The map is strictly monotone on S if v ≺ w implies that
T (v) ≺ T (w) for all v, w ∈ S. Clearly, being related is invariant under
iteration of a strongly monotone map.
Throughout this paper we shall use the North-East ordering for which the
positive cone is the first quadrant, i.e. this partial ordering is defined by
(x1 , y1 ) ¹ne (x2 , y2 )
⇔
x1 ≤ x2 and y1 ≤ y2
(5)
A map T on a nonempty set S ⊂ R2 which is monotone with respect
to the North-East ordering is called cooperative and a map monotone with
respect to the South-East ordering
(x1 , y1 ) ¹se (x2 , y2 )
⇔
x1 ≤ x2 and y1 ≥ y2
(6)
is called competitive.
If T is differentiable map on a nonempty set S, a sufficient condition for
T to be strongly monotonic with respect to the NE ordering is that the
Jacobian matrix at all points x has the sign configuration
¸
·
+ +
,
(7)
sign (JT (x)) =
+ +
provided that S is open and convex.
For x ∈ R2 , define Q` (x) for ` = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, Q1 (x) = {y ∈ R2 : x1 ≤ y1 , x2 ≤ y2 }. The (open) ball of
radius r centered at x is denoted with B(x, r). If K ⊂ R2 and r > 0,
write K + B(0, r) := {x : x = k + y for some k ∈ K and y ∈ B(0, r)}. If
x ∈ [−∞, ∞]2 is such that x ¹ y for every y in a set Y, we write x ¹ Y.
The inequality Y ¹ x is defined similarly.
The following definition is from [31].
Definition 1. Let S be a nonempty subset of R2 . A competitive map T :
S → S is said to satisfy condition (O+) if for every x, y in S, T (x) ¹ne T (y)
implies x ¹ne y, and T is said to satisfy condition (O−) if for every x, y in
S, T (x) ¹ne T (y) implies y ¹ne x.
The following theorem was proved by de Mottoni-Schiaffino [6] for the
Poincaré map of a periodic competitive Lotka-Volterra system of differential
216
A. BRETT AND M. R. S. KULENOVIĆ
equations. Smith generalized the proof to competitive and cooperative maps
[28, 29].
Theorem 4. Let S be a nonempty subset of R2 . If T is a competitive map
for which (O+) holds then for all x ∈ S, {T n (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges
to a fixed point of T . If instead (O−) holds, then for all x ∈ S, {T 2n } is
eventually componentwise monotone. If the orbit of x has compact closure
in S, then its omega limit set is either a period-two orbit or a fixed point.
It is well known that a stable period-two orbit and a stable fixed point
may coexist, see Dancer and Hess [5].
The following result is from [31], with the domain of the map specialized
to be the cartesian product of intervals of real numbers. It gives a sufficient
condition for conditions (O+) and (O−).
Theorem 5. Let R ⊂ R2 be the cartesian product of two intervals in R. Let
T : R → R be a C 1 competitive map. If T is injective and detJT (x) > 0 for
all x ∈ R then T satisfies (O+). If T is injective and detJT (x) < 0 for all
x ∈ R then T satisfies (O−).
Theorems 4 and 5 are quite applicable as we have shown in [4], in the case
of competitive systems in the plane consisting of linear fractional equations.
The next results, from [22], are useful for determining basins of attraction
of fixed points of cooperative maps. These results generalize the corresponding result for hyperbolic case from [20]. Related results have been obtained
by H. L. Smith in [28, 29, 30].
Theorem 6. Let T be a competitive map on a rectangular region R ⊂ R2 .
Let x ∈ R be a fixed point of T such that ∆ := R ∩ int (Q1 (x) ∪ Q3 (x))
is nonempty (i.e., x is not the NW or SE vertex of R), and T is strongly
competitive on ∆. Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian JT (x) of T at x has real eigenvalues λ, µ such that
0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ
is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of
the basin of attraction of x, such that C is tangential to the eigenspace E λ
at x, and C is the graph of a strictly increasing continuous function of the
first coordinate on an interval. Any endpoints of C in the interior of R are
either fixed points or minimal period-two points. In the latter case, the set
of endpoints of C is a minimal period-two orbit of T .
MONOTONE DIFFERENCE EQUATIONS
217
We shall see in Theorem 9 that the situation where the endpoints of C are
boundary points of R is of interest. The following result gives a sufficient
condition for this case.
Theorem 7. For the curve C of Theorem 6 to have endpoints in ∂R, it is
sufficient that at least one of the following conditions is satisfied.
i. The map T has no fixed points nor periodic points of minimal periodtwo in ∆.
ii. The map T has no fixed points in ∆, det JT (x) > 0, and T (x) = x
has no solutions x ∈ ∆.
iii. The map T has no points of minimal period-two in ∆, det JT (x) < 0,
and T (x) = x has no solutions x ∈ ∆.
In many cases one can expect the curve C to be smooth.
Theorem 8. Under the hypotheses of Theorem 6, suppose there exists a
neighborhood U of x in R2 such that T is of class C k on U ∪ ∆ for some
k ≥ 1, and that the Jacobian of T at each x ∈ ∆ is invertible. Then the
curve C in the conclusion of Theorem 6 is of class C k .
For maps that are strongly competitive near the fixed point, hypothesis
b. of Theorem 6 reduces just to |λ| < 1. This follows from a change of
variables [31] that allows the Perron-Frobenius Theorem to be applied to
give that at any point, the Jacobian of a strongly competitive map has two
real and distinct eigenvalues, the larger one in absolute value being positive,
and that corresponding eigenvectors may be chosen to point in the direction
of the second and first quadrant, respectively. Also, one can show that in
such case no associated eigenvector is aligned with a coordinate axis. The
next result is useful for determining basins of attraction of fixed points of
competitive maps.
Theorem 9. (A) Assume the hypotheses of Theorem 6, and let C be the
curve whose existence is guaranteed by Theorem 6. If the endpoints of C
belong to ∂R, then C separates R into two connected components, namely
W− : = {x ∈ R \ C : ∃y ∈ C with x ¹se y}
and
W+ : = {x ∈ R \ C : ∃y ∈ C with y ¹se x} ,
(8)
such that the following statements are true.
(i) W− is invariant, and dist(T n (x), Q2 (x)) → 0 as n → ∞ for every
x ∈ W− .
(ii) W+ is invariant, and dist(T n (x), Q4 (x)) → 0 as n → ∞ for every
x ∈ W+ .
218
A. BRETT AND M. R. S. KULENOVIĆ
(B) If, in addition to the hypotheses of part (A), x is an interior point of
R and T is C 2 and strongly competitive in a neighborhood of x, then T has
no periodic points in the boundary of Q1 (x) ∪ Q3 (x) except for x, and the
following statements are true.
(iii) For every x ∈ W− there exists n0 ∈ N such that T n (x) ∈ int Q2 (x)
for n ≥ n0 .
(iv) For every x ∈ W+ there exists n0 ∈ N such that T n (x) ∈ int Q4 (x)
for n ≥ n0 .
Basins of attraction of period-two solutions or period-two orbits of certain
systems or maps can be effectively treated with Theorems 6 and 9. See
[18, 19, 24] for the hyperbolic case, for the non-hyperbolic case see [22].
If T is a map on a set R and if x is a fixed point of T , the stable set W s (x)
of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set
n
x ∈ R : there exists {xn }0n=−∞ ⊂ R s.t. T (xn ) = xn+1 ,
o
x0 = x, and lim xn = x
n→−∞
W s (x)
When T is non-invertible, the set
may not be connected and made up
u
of infinitely many curves, or W (x) may not be a manifold. The following
result gives a description of the stable and unstable sets of a saddle point
of a competitive map. If the map is a diffeomorphism on R, the sets W s (x)
and W u (x) are the stable and unstable manifolds of x.
Theorem 10. In addition to the hypotheses of part (B) of Theorem 9,
suppose that µ > 1 and that the eigenspace E µ associated with µ is not a
coordinate axis. If the curve C of Theorem 6 has endpoints in ∂R, then C
is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in
R that is tangential to E µ at x and such that it is the graph of a strictly
decreasing function of the first coordinate on an interval. Any endpoints of
W u (x) in R are fixed points of T .
Remark 2. The results for cooperative systems are analogous where in
the above mentioned results competitive should be replaced by cooperative,
Q2 (x) and Q4 (x) should be replaced with Q1 (x) and Q3 (x) respectively, and
increasing should be replaced by decreasing. See [31] for a substitution that
transform competitive map into cooperative map.
The following result gives the necessary and sufficient condition for the
local stability of
xn+1 = f (xn , xn−1 , . . . , xn−k ),
n = 0, 1, . . .
when f is non-decreasing in all its arguments, see [12].
(9)
MONOTONE DIFFERENCE EQUATIONS
219
Theorem 11. Let
zn+1 =
k
X
pi zn−i ,
n = 0, 1, . . .
(10)
i=0
be the standard linearization about the equilibrium x of Eq. (9) where pi =
∂f /∂xn−i (x̄, . . . , x̄) ≥ 0, i = 0, . . . , k. Then the equilibrium x of Eq. (9) is
one of the following:
Pk
(a) locally asymptotically stable if
pi < 1,
i=0P
k
(b) non-hyperbolic and locally stable if
i=0 pi = 1,
Pk
p
>
1.
(c) unstable if
i=0 i
Remark 3. We say that f is strongly increasing in both arguments if it
is increasing, differentiable and have both partial derivatives positive in a
considered set. The connection between the theory of monotone maps and
the asymptotic behavior of Eq. (1) follows from the fact that if f is strongly
increasing, then a map associated to Eq. (1) is a cooperative map on I ×
I while the second iterate of a map associated to Eq. (1) is a strictly
cooperative map on I × I.
Set xn−1 = un and xn = vn in Eq. (1) to obtain the equivalent system
un+1 = vn
,
vn+1 = f (vn , un )
n = 0, 1, . . . .
Let F (u, v) = (v, f (v, u)). Then F maps I ×I into itself and is a cooperative
map. The second iterate T := F 2 is given by
T (u, v) = (f (v, u), f (f (v, u), v))
and it is clearly strictly cooperative on I × I.
Remark 4. The characteristic equation of Eq. (1) at an equilibrium point
(x̄, x̄):
λ2 − D1 f (x̄, x̄)λ − D2 f (x̄, x̄) = 0,
(11)
has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever
f is strictly increasing in both variables. Here Di f, i = 1, 2 denotes the
partial derivative with respect to the i-th variable. Thus the applicability
of Theorems 6 and 9 depends on the nonexistence of minimal period-two
solution.
The theory of monotone maps has been extensively developed at the level
of ordered Banach spaces and applied to many types of equations such as ordinary, partial and discrete, see [5, 6, 10, 11, 31, 32, 33, 34]. In particular, [10]
has an extensive updated bibliography of different aspects of the theory of
monotone maps. The theory of monotone discrete maps is more specialized
and so one should expect stronger results in this case. An excellent review of
A. BRETT AND M. R. S. KULENOVIĆ
220
basic results is given in [11, 26, 31]. In particular, two-dimensional discrete
maps are studied in great details and very precise results which describe the
global dynamics and the basins of attractions of equilibrium points as well
as global stable manifolds are given in [6, 18, 20, 22, 28, 29, 30, 31, 32].
In this paper we consider Eq. (1) which has several equilibrium points and
provide the sufficient conditions for all solutions to converge to an equilibrium point. More precisely, we will give sufficient conditions for the precise
description of the basins of attraction of different equilibrium points. An
application of our results gives precise description of the basis of attraction
of equations (4) and (21).
3. Main results
The following result is needed to state our main result:
Theorem 12. Consider Eq.(1) subject to the following conditions:
(C1) f ∈ C[I 2 , I] is increasing in both arguments and I is an interval
(C2) Eq.(1) has no prime period-two solutions
Then every bounded solution of Eq.(1) converges to an equilibrium.
Proof. We will check that the map F associated to Eq.(1) satisfies condition
(O−). Indeed, assume
F (x) ¹se F (y)
for any x = (u1 , v1 ), y = (u2 , v2 ), that is
v1 ≤ v2 ,
f (v1 , u1 ) ≥ f (v2 , u2 ).
In view of monotone character of f this implies u2 ≤ u1 which implies
y ¹se x. Observe that we do not require continuity of f for this proof. Thus
the conditions of Theorem 4 are satisfied and so the conclusion follows from
this theorem.
We will provide the second proof of this result under the additional assumption that f is strictly increasing, which we need for application of
Theorems 6 and 9. We check that the conditions of Theorem 5 are satisfied
for the second iterate T = F 2 of the map associated with Eq.(1).
If D1 g(u, v) and D2 g(u, v) denote the partial derivatives of a function
g(u, v) with respect to u and v, the Jacobian matrix of T is


JT (u, v) = 

D2 f (v, u)
D1 f (v, u)
D1 f (f (v, u), v) D2 f (v, u)
D1 f (f (v, u), v) D1 f (v, u)
+D2 f (f (v, u), v)
The determinant of (12) is given by
det JT (u, v) = D2 f (v, u)D2 f (f (v, u), v) > 0.


.

(12)
MONOTONE DIFFERENCE EQUATIONS
221
To check injectivity of T we set
T ((u1 , v1 )) = T ((u2 , v2 ))
which implies
f (v1 , u1 ) = f (v2 , u2 ),
f (f (v1 , u1 ), v1 ) = f (f (v2 , u2 ), v2 )
and so f (f (v1 , u1 ), v1 ) = f (f (v1 , u1 ), v2 ). By using the monotonicity of f we
conclude that v1 = v2 which in view of f (v1 , u1 ) = f (v2 , u2 ) gives u1 = u2 .
∞
Theorems 4 and 5 impliy that the subsequences {x2k }∞
k=0 and {x2k−1 }k=0
of every solution of Eq.(1) are eventually monotonic. Furthermore, every
bounded solution converges to a period-two solution and in view of (C2) to
an equilibrium.
¤
Corollary 1. Consider Eq.(1) subject to (C1) and the negative feedback
condition (3). Then every bounded solution of Eq.(1) converges to an equilibrium.
Proof. We show that the negative feedback condition (3) implies (C2), that
is, the non-existence of a prime period-two solution. If f (x, x) > x or
f (x, x) < x for all x < x̄ or x > x̄, then v > u implies that u = f (v, u) >
f (u, u) for u < x̄ and v = f (u, v) < f (v, v) for v > x̄, which shows that
Eq.(1) can not have period-two solution.
¤
Remark 5. Theorem 12 was established independently in [1] by direct
proof. We are providing the proof here for two reasons. First, we want
to show that the result follows from more general Theorem 4. Second, we
want to show that the conditions (a) and (b) of Theorem 6 are satisfied
for Eq.(1) when f is strictly increasing and the corresponding map is C 2 in
some neighborhood of the equilibrium point.
Theorem 13 and Corollary 1 are results that hold for second order difference equations and can not be extended to higher order difference equations
because Theorems 4 and 5 which are fundamental in the proof can not be
extended to higher order difference equations. Next, we will prove a slightly
less general result for Eq.(1) which can be extended to higher order difference
equations.
Theorem 13. Consider Eq.(1) subject to (C1) and the following conditions:
(C3) There exist two equilibrium points 0 ≤ x1 < x2 of Eq.(1)
(C4) Either the negative feedback condition (NFC) with respect to x1 holds
(x − x1 )(f (x, x) − x) < 0, ∀x ∈ (x1 , x2 )
or
(C5) the negative feedback condition (NFC) with respect to x2 holds
(x − x2 )(f (x, x) − x) < 0, ∀x ∈ (x1 , x2 ).
A. BRETT AND M. R. S. KULENOVIĆ
222
Then every bounded solution of Eq.(1) converges to an equilibrium. The
box (x1 , x2 )2 is a part of the basin of attraction of x1 if (C4) is satisfied or
x2 if (C5) is satisfied.
Proof. Assume that x−1 , x0 ∈ [0, x1 ] and that at least one of them is smaller
than x1 . By using the monotonicity of f we obtain
x1 = f (x0 , x−1 ) < f (x1 , x1 ) = x1 .
By using the inductive argument we can show that xn ∈ [0, x1 ), ∀n ≥ −1,
which proves that [0, x1 ] is an attracting interval. In addition f satisfies
0 < f (u, v) ≤ f (x1 , x1 ) = x1 , ∀u, v ∈ [0, x1 ].
Thus [0, x1 ) is an invariant interval and f : [0, x1 ]2 → [0, x1 ]. Consequently,
all conditions of Theorem 2 are satisfied and so we conclude that
lim xn = x1
n→∞
(13)
for every solution {xn } of Eq.(1).
A similar argument can be applied in the case where x−1 , x0 ≥ x2 and at
least one inequality is strict. In this case, if U > x2 is a number such that
xn ≤ U,
∀n ≥ −1
such that
f (u, v) ≤ f (U, U ) ≤ U
which shows that [x2 , U ] is an invariant (and attracting) interval. Thus an
application of Theorem 2 leads to
lim xn = x2 .
n→∞
(14)
Next, assume that x−1 , x0 ∈ [x1 , x2 ]. For the sake of definiteness assume
that
x1 ≤ x−1 ≤ x0 ≤ x2
where at least one inequality is strict.
By using (C2 ), we have
x1 = f (x1 , x1 ) ≤ x1 = f (x0 , x−1 ) ≤ f (x2 , x2 ) = x2 .
By using induction we can show that xn ∈ [x1 , x2 ], n ≥ −1 and so [x1 , x2 ] is
an attracting interval. Let
x1 < U = max{x−1 , x0 } < x2 .
If (C4 ) holds then
x1 < f (x1 , x1 ) < x1 = f (x0 , x−1 ) ≤ f (U, U ) < U.
That is x1 < x1 < U . Furthermore, if u, v ∈ [x1 , U ] then (C4 ) implies
x1 = f (x1 , x1 ) ≤ f (u, v) ≤ f (U, U ) < U.
MONOTONE DIFFERENCE EQUATIONS
223
Consequently, by applying Theorem 2 in an invariant interval [x̄1 , U ] we get
that the solution satisfies (13).
Assume now that x2 = x−1 > x0 > x1 . Let L = min{x−1 , x0 }. If (C5 )
holds, then we obtain
x1 = f (x0 , x−1 ) > f (L, L) > L
and by (C2 ) we have
L < f (L, L) ≤ f (u, v) ≤ f (x2 , x2 ) = x2 ,
for all u, v ∈ [L, x2 ]. Thus [L, x2 ] is an invariant interval for f .
By applying Theorem 2 we obtain that the solutions satisfy (14).
The case x2 = x−1 > x0 = x1 or x2 = x0 > x−1 = x1 can be reduced
to one of the previous cases. Namely, it is easy to show that in this case
x1 , x2 ∈ (x1 , x2 ) and so we can define L and U starting with x1 and x2 .
Now we want to consider the remaining cases for the initial conditions
such as x−1 ≤ x̄2 < x0 and prove that all bounded solutions converge to an
equilibrium. To complete this task we will use the basic results of monotone
maps, see [31], [19].
Based on our previous claims we know that the basin of attraction B(E1 )
of E1 contains Q3 (E1 ) and also Q1 (E1 ) ∩ Q3 (E2 ) if (C4 ) is satisfied while
the basin of attraction B(E2 ) contains Q1 (E2 ), and in the case when (C5 )
is satisfied, Q1 (E1 ) ∩ Q3 (E2 ).
Take any point P (x−1 , x0 ) outside of the set A = Q1 (E2 ) ∪ Q4 (E4 ) ∪
Q1 (E1 ) ∩ Q3 (E2 ). Then, one can find two points Pl ∈ Q3 (E1 ) and Pu ∈
Q1 (E2 ) such that
Pl ¹ P ¹ Pu
with respect to the North-East ordering.
By using the monotonicity of a map F associated with Eq.(1) we have
F (Pl ) ¹ F (P ) ¹ F (Pu )
and
F n (Pl ) ¹ F n (P ) ¹ F n (Pu ).
As we have seen
lim F n (Pl ) = E1 and lim F n (Pu ) = E2 ,
n→∞
n→∞
which by continuity (see [20]) implies that
lim F n (P ) ∈ Q1 (E1 ) ∩ Q3 (E2 )
n→∞
and so
lim F n (P ) = E1
n→∞
A. BRETT AND M. R. S. KULENOVIĆ
224
if (C4 ) holds, and
lim F n (P ) = E2
n→∞
if (C5 ) holds.
This means that the solution {xn } satisfies (13) when (C4 ) holds or (14)
when (C5 ) holds.
¤
Theorem 13 can be combined with Theorems 6 and 9 to give the precise
description of basins of attraction of equilibrium points of Eq.(1). Here is
our main result:
Theorem 14. Consider Eq.(1) where f is increasing function in its arguments and assume that there is no minimal period-two solution. Assume
that E1 (x1 , y1 ), E2 (x2 , y2 ), and E3 (x3 , y3 ) are three consecutive equilibrium
points in North-East ordering that satisfy
(x1 , y1 ) ¹ne (x2 , y2 ) ¹ne (x3 , y3 )
and that E1 , E3 are saddle points with the neighborhoods where f is strictly
increasing and E2 is a local attractor.
Then the basin of attraction B(E2 ) of E2 is the region between the global
stable manifolds W s (E1 ) and W s (E3 ). More precisely
n
B(E2 ) = (x, y) : ∃yu , yl : yl < y < yu ,
o
(x, yl ) ∈ W s (E1 ), (x, yu ) ∈ W s (E3 ) .
The basins of attraction B(E1 ) = W s (E1 ) and B(E3 ) = W s (E3 ) are exactly
the global stable manifolds of E1 and E3 .
Proof. In view of Theorem 3 all solutions are eventually monotonic and so
all bounded solutions are convergent to the equilibrium points. Set
n
o
B = (x, y) : ∃yu , yl : yl < y < yu , (x, yl ) ∈ W s (E1 ), (x, yu ) ∈ W s (E3 ) .
Straightforward calculation shows that the eigenspaces E λ associated with
the stable eigenvalues λ, |λ| < 1 is [1, λ] and so is not a coordinate axis.
The existence and the properties of the global stable manifolds W s (E1 ) and
W s (E3 ) are guaranteed by Theorems 6 and 9. Take (x−1 , x0 ) ∈ B. Then
there exist the points (xl−1 , xl0 ) ∈ W s (E1 ) and (xu−1 , xu0 ) ∈ W s (E3 ) such that
(xl−1 , xl0 ) ¹ (x−1 , x0 ) ¹ (xu−1 , xu0 ),
which implies
T n ((xl−1 , xl0 )) ¹ T n ((x−1 , x0 )) ¹ T n ((xu−1 , xu0 )).
MONOTONE DIFFERENCE EQUATIONS
225
By taking n → ∞ we obtain
E1 ¹ T n ((x−1 , x0 )) ¹ E3 .
In view of the uniqueness of the global stable manifold and Theorem 9 we
obtain that
lim T n ((x−1 , x0 )) = E2 ,
n→∞
which shows that B ⊂ B(E2 ).
The proof that B(E1 ) = W s (E1 ) and B(E3 ) = W s (E3 ) follows from
Theorems 9 and 10.
¤
Remark 6. The requirement that E1 and E3 are saddle points can be replaced by the requirement that at least one of them is non-hyperbolic point
that satisfies conditions (a) and (b) of Theorem 6. See Example 6. The
assumptions of Theorem 14 are of local character for all three equilibrium
points, and no strict increasing character is assumed for E2 . Even with assumptions we can characterize the basin of attraction of E2 . See Example 7.
Remark 7. Theorem 13 can be extended to the case when Eq.(1) has a
finite number of equilibrium points. Also Theorem 13 can be extended to
the case of the (k + 1)-st order difference equation (9).
Theorem 15. Consider Eq.(9) subject to the following conditions:
(C1) f ∈ C[[0, ∞)k+1 , [0, ∞)]
(C2) f (u0 , u1 , . . . , xk ) is increasing in all arguments
(C3) There exist m equilibrium points 0 ≤ x1 < x2 < . . . < xm of Eq.(9)
(C4) The negative feedback condition with respect to xi for some i ∈
{1, . . . , m} holds
(x − xi )(f (x, x, . . . , x) − x) < 0, ∀x ∈ (xi , xi+1 ).
Then every bounded solution of Eq.(1) converges to an equilibrium. The
box (xi , xi+1 )k+1 is a part of the basin of attraction of xi .
4. Examples
Here we present some examples of applications of our theorems.
Example 2. Equation
pxn + xn−1
, n = 0, 1, . . .
(15)
r + pxn + xn−1
where p, r > 0 and the initial conditions x−1 , x0 are non-negative was considered in [2]. Eq.(15) has zero equilibrium always and when p + 1 > r there
exists an additional positive equilibrium x = (p + 1 − r)/(p + 1). Clearly the
function
pu + v
, u, v ≥ 0
f (u, v) =
r + pu + v
xn+1 =
A. BRETT AND M. R. S. KULENOVIĆ
226
is strictly increasing function in both arguments and
f (u, v) =
pu + v
max{p, 1}
≤
= U,
r + pu + v
min{p, r, 1}
u, v ≥ 0.
It is easy to show that the zero equilibrium is a global attractor for p+1 < r.
Assume that p + 1 > r. Then for 0 < x < x we have
µ
¶
µ
¶
p + 1 − r − (p + 1)x
x−x
f (x, x) − x = x
= (p + 1)x
> 0,
r + (p + 1)x
r + (p + 1)x
which shows that the negative feedback condition with respect to the positive
equilibrium is satisfied. Theorem 13 implies (13) for every solution with at
least one positive initial condition.
Example 3. We give an example which is relevant to mathematical biology,
see [25] for a special case. Consider
xn+1 = f1 (xn ) + f2 (xn−1 ),
n = 0, 1, . . . ,
(16)
where f1 , f2 are continuous, increasing functions on [0, ∞). Assume that
there exists unique x > 0 such that
f1 (0) + f2 (0) = 0
and f1 (x) + f2 (x) = x.
Assume that
f1 (x) + f2 (x) > x
for any 0 < x < x. Then every positive solution of Eq. (16) will converge to
x. This result is an immediate application of Theorem 1. The special case
of this result when
fi (u) =
βi u
,
1 + bi u
βi , bi > 0 i = 1, 2,
was given in [25]. Equation
xn+1 =
β1 xn
β2 xn−1
+
1 + b1 xn 1 + b2 xn−1
(17)
has zero equilibrium always and when β1 + β2 > 1 it has also an additional
positive equilibrium point x. Indeed, the positive equilibrium satisfies
1=
β2
β1
+
= h(x).
1 + b1 x 1 + b2 x
Clearly h(u) is a decreasing function for u > 0 and
h(0) = β1 + β2 > 1,
lim h(u) = 0,
u→∞
MONOTONE DIFFERENCE EQUATIONS
227
which shows that there exists unique x > 0 such that h(x) = 1. Let us
check the negative feedback condition with respect to zero equilibrium for
function f (x, y). We have
β1 x
β2 x
β1
β2
f (x, x) − x =
+
− x = x(
+
− 1)
1 + b1 x 1 + b2 x
1 + b1 x 1 + b2 x
= x(h(x) − 1) > x(h(x) − 1) = 0 if x < x.
Theorem 13 implies
lim xn = x.
n→∞
for every solution of Eq.(17).
Example 4. We give another example which is relevant to mathematical
biology, see [7, 16, 15, 32]. Consider
pxn + xn−1
xn+1 =
, n = 0, 1, . . .
(18)
qxn + xn−1
where p, q are positive and the initial conditions x−1 , x0 are nonnegative,
and such that x−1 + x0 > 0. It has been proved in [16] that the unique
equilibrium of Eq.(18) is locally asymptotically stable if q < pq + 1 + 3p and
it was conjectured that it is globally asymptotically stable as well. Recently,
the global asymptotic stability was established in the region q < p in [18]
and [26]. Here we show how we can establish global attractivity of the
equilibrium by embedding Eq.(18) into a higher order difference equation
which is increasing in all its arguments. The embedding is performed by
one or more iterations of this equation. This method was used extensively
in a recent monograph by Camouzis and Ladas [3] to prove the boundedness
of solutions of difference equations and to find the explicit bounds for the
solutions, as well as in [12] to find the global attractivity and global stability
results for general nonlinear difference equations.
By iterating Eq.(18) once we get
xn+2 =
qx2n + xn xn−1 + p2 xn + pxn−1
= g(xn , xn−1 ), n = 0, . . .
qx2n + xn xn−1 + pqxn + qxn−1
(19)
where
qu2 + uv + p2 u + pv
qu2 + uv + pqu + qv
is decreasing in both arguments. By iterating equation (19) once more we
get an equation of order 6 where the right hand side is increasing in all its
arguments:
g(u, v) =
xn+2 = g(g(xn−2 , xn−3 ), g(xn−3 , xn−4 )) = F (xn−2 , xn−3 , xn−4 ),
(20)
for n = 0, 1, . . .. A straightforward checking shows that all three equations (18)-(20) have a unique positive equilibrium point and that F (u, v, z)
A. BRETT AND M. R. S. KULENOVIĆ
228
satisfies negative feedback condition (NFC). Thus, the unique positive equilibrium is a global attractor of all solutions of (20), and so of all solutions
of (18).
Example 5. Here we consider the following equation
1
xn+1 = (xn + xn−1 + sin(xn−1 )), n = 0, 1, . . .
(21)
2
on an interval [0, 2Kπ], where K is an integer. This equation has 2K + 1
equilibrium points x̄m = mπ, m = 0, 1, . . . , K. An immediate checking
shows that x̄2m+1 are locally asymptotically
stable, while x̄2m are saddle
√
1± 17
equilibrium points with eigenvalues λ± = 4 . The expression f (x, x)−x,
where f (x, y) = 12 (x + y + sin(y)) becomes
1
sin x < 0 if x ∈ ((2m + 1)π, (2m + 2)π).
2
This shows that the negative feedback condition is satisfied in the interval
(x̄2m+1 , x̄2m+2 ) and so the product of this interval by itself is a part of the
basin of attraction of x̄2m+1 . Similarly (x̄2m , x̄2m+1 )2 is a part of the basin
of attraction of x̄2m+1 . In particular, Theorem 3 implies that all solutions
of Eq.(21) converge to an equilibrium.
In fact, we can give precise descriptions of the basins of attraction of all
equilibrium points. Theorem 14 can be used to show that
f (x, x) − x =
B2m+1 = {(x, y) : ∃yu , yl : yl < y < yu
(x, yl ) ∈ W2m , (x, yu ) ∈ W2m+2 }
and
B2m = W2m ,
where W2m = W(x̄2m ,x̄2m ) is the global stable manifold of the equilibrium
point (x̄2m , x̄2m ). In other words, the basin of attraction of (x̄2m+1 , x̄2m+1 )
is the set of all points in the plane of initial conditions which are between
the stable manifolds of two consecutive saddle equilibrium points (x̄2m , x̄2m )
and (x̄2m+2 , x̄2m+2 ). Indeed, we only need to show that Eq.(21) does not
possess a minimal period-two solution Φ, Ψ, Φ, Ψ, . . .. Such a solution would
satisfy the system of equations
sin(Φ) = Φ − Ψ,
sin(Ψ) = Ψ − Φ
and so
Ψ = Φ − sin(Φ) = G(Φ),
Φ = Ψ − sin(Ψ) = G(Ψ),
which implies that both Φ and Ψ are period-two points of an increasing
function G(x) = x − sin(x), which is impossible.
These results are summarized in the plot in Figure 2.
MONOTONE DIFFERENCE EQUATIONS
229
W4
10
(x̄3 , x̄3 )
B3
(x̄2 , x̄2 )
5
W2
(x̄1 , x̄1 )
0
(x̄0 , x̄0 )
W0
B1
-5
-5
0
5
10
Figure 2: Figure shows the basins of attraction of Eq.(21). Figure was
generated by Dynamica 3 [17].
Example 6. Here we consider the following equation
1
xn+1 = (xn + xn−1 + (sin(xn−1 ))2 ), n = 0, 1, . . .
(22)
2
on an interval [0, 2Kπ], where K is an integer. This equation has 2K + 1
equilibrium points x̄m = mπ, m = 0, 1, . . . , K. An immediate checking
shows that x̄m are non-hyperbolic equilibrium points with eigenvalues equal
to 1 and −1/2. The expression f (x, x) − x, where f (x, y) = 12 (x + y +
(sin(y))2 ) becomes
1
f (x, x) − x = (sin x)2 > 0 for every x 6= x̄m .
2
This shows that the negative feedback condition is satisfied in the interval
(x̄m−1 , x̄m ) and so the product of this interval by itself is a part of the basin
of attraction of x̄m . In particular, Theorem 3 implies that all solutions of
Eq.(22) converge to an equilibrium. In fact, we can give precise descriptions
of the basins of attraction of all equilibrium points. Theorem 14 can be used
A. BRETT AND M. R. S. KULENOVIĆ
230
to show that
Bm = {(x, y) : ∃yu , yl : yl < y < yu
(x, yl ) ∈ Wm , (x, yu ) ∈ Wm+1 }
where Wm = W(x̄m ,x̄m ) is the global stable manifold of the equilibrium point
(x̄m , x̄m ). In other words, the basin of attraction of (x̄m , x̄m ) is the set of
all points in the plane of initial conditions which are between the stable
manifolds of two consecutive non-hyperbolic equilibrium points (x̄m , x̄m )
and (x̄m+1 , x̄m+1 ).
Indeed, we only need to show that Eq.(22) does not possess a minimal
period-two solution Φ, Ψ, Φ, Ψ, . . ..
A minimal period-two solution would satisfy the system of equations
(sin(Φ))2 = Φ − Ψ,
(sin(Ψ))2 = Ψ − Φ
and so
Ψ = Φ − (sin(Φ))2 = H(Φ),
Φ = Ψ − (sin(Ψ))2 = H(Ψ),
which implies that both Φ and Ψ are period-two points of an increasing
function H(x) = x − (sin(x))2 , which is impossible. A direct calculation
shows that the eigenvector which corresponds to the smaller eigenvalue λ =
−1/2 is [2, −1].
Example 7. Here we consider the following equation
xn+1 = x3n + x3n−1 ,
n = 0, 1, . . .
(23)
√
on√an interval (−∞,
√ ∞). This equation has 3 equilibrium points −1/ 2, 0,
1/ 2, where ±1/ 2 are saddle points and 0 is a local attractor, with both
roots of characteristic equation equal to 0. Thus, the function f (u, v) = u3 +
v 3 is strictly increasing for all values u 6= 0, v 6= 0. All conditions of Theorem
14 are satisfied with respect to two saddle equilibrium points (period-two
solution clearly does
guarantee
the√existence of two stable
√ not√exist), which
√
s
s
manifolds W ((1/ 2, 1/ 2)) and W ((−1/ 2, −1/ 2)), which are continuous, decreasing curves√that extend
indefinitely. In
√
√ view of√Theorem 9. the region between W s ((1/ 2, 1/ 2)) and W s ((−1/ 2, −1/ 2))√is invariant,
as
√
s ((1/ 2, 1/ 2)) and
well as the regions of points
which
are
north-east
of
W
√
√
south-west of W s ((−1/ 2, −1/ 2))√and the
√ basin of attraction
√ of (0,√0) is
precisely the region between W s ((1/ 2, 1/ 2))√and W√s ((−1/ 2, −1/ 2)).
All solutions which starts north-east of W s ((1/ √2, 1/ √2)) goes to ∞ while
all solutions which starts south-west of W s ((1/ 2, 1/ 2)) goes to −∞. In
view of Theorem 3 all solutions are eventually monotonic. See Figure 3.
For example, x−1 > 1, x0 ≥ 0 implies x1 = x30 + x3−1 ≥ x3−1 = x2−1 x−1 >
x−1 and x2 = x31 + x30 > x3−1 + x30 = x1 . Similarly we can prove that
{xn }n≥1 is strictly increasing and so is asymptotic to ∞. Similar reasoning
MONOTONE DIFFERENCE EQUATIONS
231
holds for x−1 ≥ 0, x0 > 1. The initial point (1, 1) generates the solution
{1, 1, 2, 9, 737, . . .}.
2
1
0
-1
-2
-2
-1
0
1
2
Figure 3: Figure shows the basins of attraction of Eq.(23). Figure was
generated by Dynamica 3 [17].
Acknowledgment. The authors are grateful to the anonymous referee for
number of insightful comments that improved the statements and exposition
of results, and in particular, for suggestion to include Example 7.
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(Received: June 25, 20098)
(Revised: October 31, 2009)
A. Brett and M. R. S. Kulenović
Department of Mathematics
University of Rhode Island
Kingston, Rhode Island 02881-0816
USA
E–mail: [email protected]